Effective Conductivity of Densely Packed Disks and Energy of Graphs

Nawalaniec, Wojciech; Necka, Katarzyna; Mityushev, Vladimir

doi:10.3390/math8122161

Cited by 4 publications

(6 citation statements)

References 26 publications

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“…The evident modification to periodic structures can be taken over. We use the term "Delaunay graph" for the constructed graph Γ following [16]. The corresponding triangulation slightly differs from commonly used Delaunay triangulation in degenerate cases.…”

Section: Structural Approximation In R Dmentioning

confidence: 99%

“…Theorem 1 justifies the theory of structural approximation [14,15] on the basis of a mesoscopic discretization when edges and vertices of graph Γ correspond to inclusions. This theorem was applied to the systematic study of densely packed deterministic composites [26] with given locations of inclusions, and to random two-dimensional composites [16]. Delaunay graph Γ determines summation (17) in energy (18).…”

Section: Structural Approximation In R Dmentioning

confidence: 99%

“…This paper is devoted to extension of previous results [16,23,24] to R d , and is partially presented in arXiv:1412.7527. Section 2 presents structural approximation theory in R d .…”

Section: Introductionmentioning

confidence: 99%

“…Recent results in structural approximations [14,15] showed that densely packed composites including random media [16] can be investigated by means of the functional associated to discrete energy. It has the following structure: min…”

Section: Introductionmentioning

confidence: 99%

“…Some results on 2D structural approximations were established in the papers outlined below. Random doubly periodic graphs were studied and applied to the homogenization problem in [16]. The minimization of periodic Functional (1) with respect to its geometry was applied to pattern formation in [23,24].…”

Section: Introductionmentioning

confidence: 99%

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Optimal Random Packing of Spheres and Extremal Effective Conductivity

Mityushev

Zhunussova

2021

Symmetry

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A close relation between the optimal packing of spheres in Rd and minimal energy E (effective conductivity) of composites with ideally conducting spherical inclusions is established. The location of inclusions of the optimal-design problem yields the optimal packing of inclusions. The geometrical-packing and physical-conductivity problems are stated in a periodic toroidal d-dimensional space with an arbitrarily fixed number n of nonoverlapping spheres per periodicity cell. Energy E depends on Voronoi tessellation (Delaunay graph) associated with the centers of spheres ak (k=1,2,…,n). All Delaunay graphs are divided into classes of isomorphic periodic graphs. For any fixed n, the number of such classes is finite. Energy E is estimated in the framework of structural approximations and reduced to the study of an elementary function of n variables. The minimum of E over locations of spheres is attained at the optimal packing within a fixed class of graphs. The optimal-packing location is unique within a fixed class up to translations and can be found from linear algebraic equations. Such an approach is useful for random optimal packing where an initial location of balls is randomly chosen; hence, a class of graphs is fixed and can dynamically change following prescribed packing rules. A finite algorithm for any fixed n is constructed to determine the optimal random packing of spheres in Rd.

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Section: Structural Approximation In R Dmentioning

confidence: 99%

Section: Structural Approximation In R Dmentioning

confidence: 99%

“…This paper is devoted to extension of previous results [16,23,24] to R d , and is partially presented in arXiv:1412.7527. Section 2 presents structural approximation theory in R d .…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Optimal Random Packing of Spheres and Extremal Effective Conductivity

Mityushev

Zhunussova

2021

Symmetry

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Singular Perturbation Problems in Periodic Domains

Riva

Cristoforis

Musolino

2021

Singularly Perturbed Boundary Value Problems

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Infiltration and permeability of porous ceramics simulated by random networks

Mityushev,

Rylko,

Dolata

et al. 2024

Arch. Civ. Mech. Eng.

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A network computational model for a 3D ceramic structure is developed. The model is applied to study the impact of geometric and material parameters of structure on the liquid metal flow through random porous ceramic medium in pressure infiltration processes. The characteristic geometric features of the ceramic structure favorable for liquid metal flow during the infiltration process are determined. The method of structural approximation and constructive homogenization are applied, and the discrete stationary Stokes equations on random graphs are considered. This approach gives a robust algorithm to determine the macroscopic permeability K of interpenetrating phases. The dependencies of K on the distribution of connections (windows) between the cells (inclusions) are derived. The numerical simulations demonstrate that the permeability K does not depend on the scaled distribution sizes of windows. This implies that K is proportional to the mean value of the window areas. The considered model takes into account a random complex structure of 3D ceramic. Hence, it complements the previous study on the local transport properties of tubes (windows) connecting the cells.

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Effective Conductivity of Densely Packed Disks and Energy of Graphs

Cited by 4 publications

References 26 publications

Optimal Random Packing of Spheres and Extremal Effective Conductivity

Optimal Random Packing of Spheres and Extremal Effective Conductivity

Singular Perturbation Problems in Periodic Domains

Infiltration and permeability of porous ceramics simulated by random networks

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